OA-Use+the+four+operations+with+whole+numbers+to+solve+problems

4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.* 4.OA.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. || ===‍‍‍‍‍‍Anchor Standard/Mathematical Practice(s) === MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. || Use technology tools and skills to reinforce classroom concepts and activities. || ===‍‍‍‍‍‍Revised Bloom's Level of thinking === Remembering Understanding Applying Analyzing <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Evaluating <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Creating || ===<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Learning Target/Task Analysis ===
 * ===<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Common Core Standards ===
 * ===<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Information Technology Standard ===

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">quantity is being multiplied and which number tells how many times. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Students should be given opportunities to write and identify equations and statements for multiplicative <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">comparisons.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Example:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">5 x 8 = 40. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Sally is five years old. Her mom is eight times older. How old is Sally’s Mom?

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">5 x 5 = 25

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Sally has five times as many pencils as Mary. If Sally has 5 pencils, how many does Mary have? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">This standard calls for students to translate comparative situations into equations with an unknown and solve. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Students need many opportunities to solve contextual problems. Refer to Glossary, Table 2(page 89) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">For more examples (table included at the end of this document for your convenience) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Examples: <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Unknown Product: A blue scarf costs $3. A red scarf costs 6 times as much. How much does the red scarf cost? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">(3 x 6 = p). <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Group Size Unknown: A book costs $18. That is 3 times more than a DVD. How much does a DVD cost? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">(18 ÷ p = 3 or 3 x p = 18). <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Number of Groups Unknown: A red scarf costs $18. A blue scarf costs $6. How many times as much does the red <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">scarf cost compared to the blue scarf? (18 ÷ 6 = p or 6 x p = 18).

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">When distinguishing multiplicative comparison from additive comparison, students should note that

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•additive comparisons focus on the difference between two quantities (e.g., Deb has 3 apples and Karen <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">has 5 apples. How many more apples does Karen have?). A simple way to remember this is, “How many <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">more?” <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•multiplicative comparisons focus on comparing two quantities by showing that one quantity is a specified number of times larger or smaller than the other (e.g., Deb ran 3 miles. Karen ran 5 times as many miles <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">as Deb. How many miles did Karen run?). A simple way to remember this is “How many times as much?” or “How many times as many?”

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">The focus in this standard is to have students use and discuss various strategies. It refers to estimation strategies,

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be structured <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">so that all acceptable estimation strategies will arrive at a reasonable answer. Students need many opportunities <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">solving multistep story problems using all four operations.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Example:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">third day. How many miles did they travel total?

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Some typical estimation strategies for this problem:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Student 1 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I first thought about 267 and 34. I noticed that their sum is about 300. Then I knew that 194 is close to 200. When I put 300 and 200 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">together, I get 500. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Student 2 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I first thought about 194. It is really close to 200. I also have 2 hundreds in 267. That gives me a total of 4 hundreds. Then I have 67 in 267 and the 34. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">When I put 67 and 34 together that is really close to 100. When I add that hundred to the 4 hundreds that I already had, I end up with 500. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Student 3 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I rounded 267 to 300. I rounded 194 to 200. I rounded 34 to 30. When I added 300, 200 and 30, I know my answer will be about 530.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">The assessment of estimation strategies should only have one reasonable answer (500 or 530), or a range

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">(between 500 and 550). Problems will be structured so that all acceptable estimation strategies will arrive at a <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">reasonable answer.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Example 2:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Your class is collecting bottled water for a service project. The goal is to collect 300 bottles of water. On the first <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">day, Max brings in 3 packs with 6 bottles in each container. Sarah wheels in 6 packs with 6 bottles in each <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">container. About how many bottles of water still need to be collected?

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">This standard references interpreting remainders. Remainders should be put into context for interpretation.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">ways to address remainders: <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•Remain as a left over <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•Partitioned into fractions or decimals <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•Discarded leaving only the whole number answer <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•Increase the whole number answer up one <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•Round to the nearest whole number for an approximate result

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Example:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Write different word problems involving 44 / 6 = ? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">where the answers are best represented as:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem A: 7

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem B: 7 r 2

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem C: 8

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem D: 7 or 8

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem E: 7 2/6

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">possible solutions:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">A: 7. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches did she fill? 44 ÷ 6 = p; p = 7 r 2. Mary can fill 7 pouches completely.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">B: 7 r 2. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches could she fill and how many pencils would she have left? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">44 ÷ 6 = p; p = 7 r 2; Mary can fill 7 pouches and have 2 left over.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">C. 8.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Mary had 44 pencils. Six pencils fit into each of her pencil pouches. What would the fewest number of pouches she would need in order to hold all of her pencils? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">44 ÷ 6 = p; p = 7 r 2; Mary can needs 8 pouches to hold all of the pencils.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">D: 7 or 8. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Mary had 44 pencils. She divided them equally among her friends before giving one of the leftovers to each of her friends. How many pencils could her friends have received? 44 ÷ 6 = p; p = 7 r 2; Some of her friends received 7 pencils. Two friends received 8 pencils.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Problem

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">E: 7 2/6 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Mary had 44 pencils and put six pencils in each pouch. What fraction represents the number of pouches that Mary filled? 44 ÷ 6 = p; p = 7 2/6

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Example:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">There are 128 students going on a field trip. If each bus held 30 students, how many buses are needed? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">(128 ÷ 30 = b; b = 4 R 8; They will need 5 buses because 4 busses would not hold all of the students). <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Students need to realize in problems, such as the example above, that an extra bus is needed for the 8 students <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">that are left over.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Estimation skills include identifying when estimation is appropriate, determining the level of accuracy needed,

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">situations using various estimation strategies. Estimation strategies include, but are not limited to: <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•front-end estimation with adjusting (using the highest place value and estimating from the front end, <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">making adjustments to the estimate by taking into account the remaining amounts), <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•clustering around an average (when the values are close together an average value is selected and <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">multiplied by the number of values to determine an estimate), <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•rounding and adjusting (students round down or round up and then adjust their estimate depending on <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">how much the rounding affected the original values), <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•using friendly or compatible numbers such as factors (students seek to fit numbers together - e.g., <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">rounding to factors and grouping numbers together that have round sums like 100 or 1000), <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">•using benchmark numbers that are easy to compute (students select close whole numbers for fractions or decimals to determine an estimate). <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍I can... <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I can write an equation for my multiplication word problem. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I can use algebraic thinking to solve word problems involving multiplication and division. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I can use the 4 operations to solve multi step word problems containing whole numbers. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I can interpret the remainder in a division word problem. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">I can make sure my answer makes sense.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Essential Vocabulary
===<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">multiplication equation, multiply, divide, unknown, algebraic thinking, equation, remainders, reasonableness, mental computation, estimation strategies, rounding ===

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Sample Assessments
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">ticket out the door

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Evaluate 8 x 6, 5 x 9, 7 x 3, etc. Write the equations for each multiplication problem using the Commutative Property of Multiplication (e.g., 8 x 6 = 6 x 8). Write and then solve the given equation using another method. Use a verbal statement to explain the chosen method.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Pedro has invited 8 of his friends to a summer party. He asked each of them to bring 7 pieces of candy. Create a representation of the total number of candy pieces the friends will share. Write the equation that represents the illustration you created. Solve for the answer.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Over the summer, Raul read 8 books. Natalia read 4 times as many books. How many books did Natalia read? Draw a picture or create a model of the problem, write an equation with a symbol for the unknown variable, and solve.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Intervention:
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">[] multiplication facts review

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Use base ten blocks

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Breaking apart-decomposing <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">25 x 8 = <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">20 x 8 = 160 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">5 x 8 = 40 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">160+40=200

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Array Model/Area Model <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">17 x 5 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">50 + 35 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">85 <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Lattice Math <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Choice Boards <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Use playing cards for multiplication war <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Post words with visuals on word wall and in vocabulary notebooks. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Use Frayer Models and foldables
 * || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">10 || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">7 ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">5 || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">50 || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">35 ||

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Enrichment:
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Students create and solve real world word problems. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Learning Logs <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">In small groups create a board game involving multiplication

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Instructional Resources
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">http//www.quia.com/mathjourney.html <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">@http://www.coolmath-games.com/numbermonster/mult25.htm <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">@http://classroom.jc-schools.net/basic/mathasmd.html ===<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">‍‍‍‍‍‍Notes and Additional Information ===